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\title{A combinatorial Chevalley formula for semi-infinite flag manifolds and its applications}

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\author{Cristian Lenart\thanks{\href{mailto:clenart@albany.edu}{clenart@albany.edu}. C. Lenart was partially supported by the NSF grants DMS-1362627 and DMS-1855592.}\addressmark{1}, Satoshi Naito\thanks{\href{mailto:naito@math.titech.ac.jp}{naito@math.titech.ac.jp}. S. Naito was partially supported by JSPS Grant-in-Aid for Scientific Research (B) 16H03920.}\addressmark{2}, \and Daisuke Sagaki\thanks{\href{mailto:sagaki@math.tsukuba.ac.jp}{sagaki@math.tsukuba.ac.jp}.
D. Sagaki was partially supported by JSPS Grant-in-Aid 
for Scientific Research (C) 15K04803 and 19K03415.}\addressmark{3}}

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\address{\addressmark{1}Dept. of Mathematics, State Univ. of New York at Albany, Albany, NY, 12222, USA \\
\addressmark{2}Dept. of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan \\
\addressmark{3}Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan}

\received{November 20, 2020}

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\keywords{semi-infinite flag manifold, Chevalley formula, quantum Bruhat graph, quantum LS paths, quantum alcove model.}

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\abstract{We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum $K$-theory $QK_{T}(G/B)$ of the flag manifold $G/B$; this has been a longstanding conjecture. We also discuss the Chevalley formula for partial flag manifolds $G/P$. Moreover, in type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivariant) quantum $K$-theory $QK(SL_{n}/B)$.}

\resume{Nous donnons une formule combinatoire de Chevalley pour un poids arbitraire, dans la $K$-th\'eorie \'equivariante des vari\'et\'es de drapeau semi-infinies, exprim\'ee en termes du mod\`ele des alc\^oves quantique. En tant qu'application, nous prouvons la formule de Chevalley pour les poids fondamentaux anti-dominantes dans la (petite) $K$-th\'eorie quantique \'equivariante $QK_{T}(G/B)$ des vari\'et\'es de drapeau $G/B$; c'\'etait une conjecture depuis longtemps. Nous discutons \'egalement de la formule plus g\'en\'erale de Chevalley pour les vari\'et\'es de drapeau partielles $G/P$. De plus, dans le type $A_{n-1}$, nous montrons que les polyn\^omes de Grothendieck quantiques repr\'esentent bien les classes de Schubert dans la $K$-th\'eorie quantique (non-\'equivariante) $QK(SL_{n}/B)$.}%sur la structure multiplicative de $QK_{T}(G/B)$. 

\maketitle

\section{Introduction}

This paper is concerned with a geometric application of the combinatorial model known as the {\em quantum alcove model}, introduced in \cite{lalgam}. Its precursor, the alcove model of the first author and Postnikov, was used to uniformly describe the {\em Chevalley formula} in the equivariant $K$-theory of flag manifolds $G/B$~\cite{LP}. Also, the quantum alcove model was used to uniformly describe certain crystals of affine Lie algebras (single-column {\em Kirillov--Reshetikhin crystals}) and {\em Macdonald polynomials} specialized at $t=0$ \cite{lnsumk2}. The objects of the quantum alcove model (indexing the crystal vertices and the terms of Macdonald polynomials) are paths in the {\em quantum Bruhat graph} on the Weyl group, introduced by Brenti-Fomin-Postnikov. In this paper we complete the above picture, by extending to the quantum alcove model the geometric application of the alcove model, namely the $K$-theory Chevalley formula. 

To achieve our goal, we need to consider the so-called {\em semi-infinite flag manifold} $\QQ$. 
%Let $\mathbf{Q}_{G}^{\mathrm{rat}}$ denote the full semi-infinite flag manifold,
%which is the reduced ind-scheme whose set of $\mathbb{C}$-valued points is $G(\mathbb{C}(\hspace{-2pt}(z)\hspace{-2pt}))/(T \cdot N(\mathbb{C}(\hspace{-2pt}(z)\hspace{-2pt})))$,
%where $G$ is a simply-connected simple algebraic group over $\mathbb{C}$, with Borel subgroup $B = T N$, $T$ maximal torus and $N$ unipotent radical.
%In this paper, we concentrate on its semi-infinite Schubert subvariety $\QQ := \QQ(e) \subset \mathbf{Q}_{G}^{\mathrm{rat}}$ associated to the identity element $e$ of the affine Weyl group $\Waf = W \ltimes Q^{\vee}$, which we also call the semi-infinite flag manifold.
We give a Chevalley formula for an arbitrary weight in the $T \times \mathbb{C}^{*}$-equivariant $K$-group $K_{T\times{\mathbb C}^*}(\QQ)$ of $\QQ$, which is described in terms of the quantum alcove model.
In~\cite{knsekt} and \cite{nospcf}, the Chevalley formulas for $K_{T\times{\mathbb C}^*}(\QQ)$ were originally given in terms of the {\em quantum LS path model} in the case of a dominant and an anti-dominant weight, respectively.
For a general (not dominant nor anti-dominant) weight, there is no quantum LS path model, but there is a quantum alcove model.
Hence, in order to obtain a Chevalley formula for an arbitrary weight, we first need to translate the formulas above to the quantum alcove model by using the weight-preserving bijection between the two models given by Proposition~\ref{qam2qls}.
Based on these translated formulas (Theorems~\ref{chevdomqam} and \ref{corantidom}), we obtain a Chevalley formula (Theorem~\ref{genchev}) for an arbitrary weight.% by using elaborate combinatorics of the quantum alcove model.

The study of the equivariant $K$-group of semi-infinite flag manifolds was started in~\cite{knsekt}. A breakthrough in this study is \cite{kat1} (see also \cite{kat2}), in which Kato established a $\mathbb{Z}[P]$-module isomorphism from the (small) $T$-equivariant quantum $K$-theory $QK_{T}(G/B)$ of the finite-dimensional flag manifold $G/B$ onto (a version of) the $T$-equivariant $K$-group $K_{T}^{\prime}(\QQ)$ of $\QQ$; here $P$ is the weight lattice generated by the fundamental weights $\varpi_{i}$, $i \in I$.
Here we should mention that in \cite{kat1}, he also established a $\bZ[P]$-module embedding of (a certain localization of) the $T$-equivariant $K$-group of the affine Grassmannian into the $T$-equivariant $K$-group of the full semi-infinite flag manifold $\mathbf{Q}_{G}^{\mathrm{rat}}$, which is a certain inductive limit of copies of $K_{T}(\QQ)$, thus verifying a conjectural $K$-theoretic generalization of Peterson's isomorphism proposed by Lam-Li-Mihalcea-Shimozono (\cite{LLMS}).
%Here, $K_{T}^{\prime}(\QQ)$ is a $\mathbb{C}[P]$-module having a basis of semi-infinite Schubert classes $\OQG{x}$ for $x \in \Wafp := W \times Q^{\vee,+} \subset \Waf$, and the $\mathbb{C}[P]$-module 
The isomorphism above sends each (opposite) Schubert class in $QK_{T}(G/B)$ to the corresponding semi-infinite Schubert class in $K_{T}^{\prime}(\QQ)$; moreover, it respects the quantum multiplication in $QK_{T}(G/B)$ with the class of the line bundle associated to an anti-dominant fundamental weight and the tensor product in $K_{T}^{\prime}(\QQ)$ with the class of the line bundle associated to the corresponding anti-dominant fundamental weight. Based on this result, a longstanding conjecture on the multiplicative structure of $QK_{T}(G/B)$, i.e., the Chevalley formula (Theorem~\ref{qkchev}) for anti-dominant fundamental weights $- \varpi_{k}$, $k \in I$, for $QK_{T}(G/B)$ is proved by our Chevalley formula for $K_{T \times \mathbb{C}^{*}}(\QQ)$ specialized to $q = 1$. In Section~\ref{partial}, we also discuss the quantum $K$-theory Chevalley formula for partial flag manifolds $G/P$.

As an application of our quantum $K$-theory Chevalley formula, we prove an important conjecture for the non-equivariant quantum $K$-theory $QK(SL_{n}/B)$ of the type $A_{n-1}$ flag manifold (Theorem~\ref{qgroth}):
the {\em quantum Grothendieck polynomials}, introduced in \cite{lamqgp}, represent Schubert classes in $QK(SL_{n}/B)$. Thus, we generalize  the results of~\cite{fgpqsp}, where the {\em quantum Schubert polynomials} are constructed as representatives for Schubert classes in the quantum cohomology of $SL_{n}/B$.  %since our Chevalley formula for anti-dominant fundamental weights $- \varpi_{i}$, $i \in I$, determines the multiplicative structure of $QK(SL_{r+1}/B)$.
Therefore, we can use quantum Grothendieck polynomials to compute any structure constant in $QK(SL_{n}/B)$ (with respect to the Schubert basis);  
indeed, we just need to expand their products in the basis they form, which is done by \cite[Algorithm~3.28]{lamqgp}, see \cite[Example~7.4]{lamqgp}. This is important, since computing even simple products in quantum $K$-theory is notoriously difficult.

\section{Background on the combinatorial models}\label{backg}

\subsection{Root systems} Let $\mathfrak{g}$ be a complex simple Lie algebra, and $\mathfrak{h}$ a Cartan subalgebra.  Let $\Phi\subset \mathfrak{h}^*$ be the corresponding irreducible \textit{root system}, $\mathfrak{h}^*_\mathbb{R}$ the real span of the roots, and $\Phi^{+}\subset\Phi$ the set of positive roots. Given $\alpha\in\Phi$, we let $\sgn(\alpha)$ be 1 or $-1$ depending on $\alpha$ being positive or negative, and $|\alpha|:=\sgn(\alpha)\alpha$. 
Let  $\rho:=\frac{1}{2}(\sum_{\alpha\in\Phi^+}\alpha)$.
  Let $\theta$ be the highest root, and $\alpha_i\in\Phi^+$ the \textit{simple roots}, for $i$ in an indexing set $I$. We denote $\langle\cdot,\cdot\rangle$ the nondegenerate scalar product on $\mathfrak{h}^*_{\mathbb{R}}$ induced by the Killing form.  Given $\alpha\in\Phi$, we consider the \textit{coroot} $\alpha^{\vee}$ and reflection $s_{\alpha}$. The root and coroot lattices are denoted by $Q$ and $Q^\vee$, as usual, while the positive part of the coroot lattice is denoted by $Q^{\vee,+}$. The \textit{weight lattice} $P$ is generated by the \textit{fundamental weights} $\varpi_i$, for $i\in I$. Let $P^+$ be the set of \textit{dominant weights}. 

Let $W$ be the \textit{Weyl group}, with length function $\ell(\cdot)$ and longest element $w_\circ$.  The \textit{Bruhat order} on $W$ is defined by its covers $w\lessdot ws_{\alpha},$ for $\ell(ws_\alpha) = \ell(w)+1$, where $\alpha\in\Phi^+$. 

Given $\alpha\in\Phi$ and $k\in\mathbb{Z}$, we denote by $s_{\alpha,k}$ the reflection in the affine hyperplane $H_{\alpha,k}:=\{\lambda\in\mathfrak{h}^*_{\mathbb{R}} \mid \langle\lambda,\alpha^{\vee}\rangle = k\}$. 
These reflections generate the \textit{affine Weyl group} $\Waf=W\ltimes Q^\vee$ for the \textit{dual root system} $\Phi^{\vee}$. The hyperplanes $H_{\alpha,k}$ divide the vector space $\mathfrak{h}^*_{\mathbb{R}}$ into open regions, called \textit{alcoves}.  The \textit{fundamental alcove} is denoted by $A_{\circ}$. 

 The \textit{quantum Bruhat graph} ${\QB}(W)$ on $W$ is defined by adding downward (quantum) edges, denoted $w \triangleleft ws_\alpha$, to the covers of the Bruhat order, i.e., the edges of ${\QB}(W)$ are:
$$w\xrightarrow{\alpha}ws_\alpha\hspace{8pt}\text{if}\hspace{6pt} w\lessdot ws_\alpha\hspace{6pt}\mbox{or}\hspace{6pt} \ell(ws_{\alpha}) =\ell(w) - 2\langle\rho,\alpha^{\vee}\rangle + 1\,,\hspace{8pt}\mbox{where $\alpha\in\Phi^+$}\,.$$\label{quantum bruhat graph eq}
We define the {\em weight} of an edge $w \overset{\alpha}\longrightarrow {ws_\alpha}$ to be
either $\alpha^\vee$ or $0$, depending on whether it is a quantum edge or not, respectively. 
Then the weight of a directed path is 
the sum of the weights of its edges. It turns out that the weight of a shortest directed path from $v$ to $w$ is independent of the mentioned path, so we will denote it by $\wt(w\Rightarrow v)$; see~\cite{lnsumk2}. 

For the remainder of this section, we fix $\lambda\in P^+$. Let $W_J$ be the stabilizer of $\lambda$, as a parabolic subgroup with $J\subset I$ and root system $\Phi_J$. We denote the set of minimum-length coset representatives for $W/W_J$ by $W^J$, and the minimum-length coset representative of $wW_J$ by $\mcr{w}$. We consider the {\em parabolic quantum Bruhat graph} on $W^J$, denoted by $\QB(W^J)$; this generalizes $\QB(W)$, see~\cite{lnsumk1}. Its directed edges are labeled by $\alpha\in \Phi^+\setminus \Phi_J^+$. The upward edges are the covers of the Bruhat order on $W^J$, while the downward (quantum) edges $w \overset{\alpha}{\longrightarrow} \mcr{ws_\alpha}$ are given by the condition $\ell(\mcr{ws_\alpha})=\ell(w)-2\langle \rho-\rho_J, \alpha^\vee \rangle+1$. 
Given a rational number $b$, we define $\QB_{b\lambda}(W^J)$ to be the subgraph of 
 $\QB(W^{J})$ with 
the same vertex set but having only the edges with labels $\alpha$ satisfying $b\langle\lambda,\alpha^\vee\rangle\in{\mathbb Z}$. 


We now recall the quantum Bruhat graph analogue of a certain lift from $W/W_J$ to $W$ which was previously defined by Deodhar. Let $\ell(w \Rightarrow x)$ denote the length of the shortest path from $w$ to $x$ in $\QB(W)$. It was shown in \cite{lnsumk1} that, given $v,w\in W$, there exists a unique element $x \in vW_{J}$ such that 
$\ell(w \Rightarrow x)$ attains its minimum value as a function of $x\in vW_J$. For reasons explained in \cite{lnsumk1}, we denote the unique element by $\min(vW_J,\preceq_w)$, and call it a {\em quantum Deodhar lift}. 


\subsection{Quantum LS paths}\label{sec:qls}

\begin{dfn}[\cite{lnsumk2}]\label{defls} A {\em quantum LS path} $\eta\in\QLS(\lambda)$, for $\lambda\in P^+$, is given by two sequences
\begin{equation}\label{E:lsste}
(0 = b_1<b_2<b_3<\cdots<b_t<b_{t+1}=1)\,;\;\; \qquad(\phi(\eta)=\sigma_1,\,\sigma_{2},\,\ldots,\,\sigma_t=\iota(\eta))\,,
\end{equation}
where $b_k\in \mathbb{Q}$, $\sigma_k\in W^J$, and there is a directed path in $\QB_{b_k\lambda}(W^J)$ from $\sigma_{k-1}$ to $\sigma_k$, for each $k=2,\ldots,t$. The elements $\sigma_k$ are called the {\em directions} of $\eta$, while $\iota(\eta)$ and $\phi(\eta)$ are the {\em initial} and {\em final directions}, respectively. 
\end{dfn}

This data encodes the sequence of vectors 
$v_t:=(b_{t+1}-b_t)\sigma_t\lambda,\,\dots,\,v_2:=(b_3-b_2)\sigma_2\lambda,\,
 v_1:=(b_2-b_1)\sigma_1\lambda$.
 We can view $\eta\in\QLS(\lambda)$ as a piecewise-linear 
path given by the sequence of points 
%
$0,\,v_t,\,v_{t-1}+v_t,\,\dots,\,v_1+\cdots+v_t$.
%
The endpoint of the path, also called its weight, is $\wt(\eta):=\eta(1)=v_1+\cdots+v_t$. 
%
Given $w\in W$, we define the {\em initial direction of $\eta$ with respect to $w$} as $\iota({\eta},{w}):=w_t \in W$, where the sequence $(w_k)$ is calculated by the following recursive formula: 
$w_0:=w$, $w_k:=\min(\sigma_kW_J,\preceq_{w_{k-1}})$ for $k=1,\ldots,t$.
%
Also, we set $\xi(\eta,w):= \sum_{k=1}^{t} \wt ({w}_{k-1} \Rightarrow {w}_{k})$ and 
$\Deg_{w}(\eta):= - \sum_{k=1}^{t} (1-b_k) \langle{\lambda},{\wt({w}_{k-1} \Rightarrow {w}_{k})}\rangle$. 

\subsection{The quantum alcove model}\label{sec:qam}

We say that two alcoves are {adjacent} if they are distinct and have a common wall. Given a pair
of adjacent alcoves $A$ and $B$, we write $A \xrightarrow{\beta}  B$  for $\beta\in\Phi$ if the
common wall is orthogonal to $\beta$ and $\beta$ points in the direction from $A$ to $B$. 

\begin{dfn}[\cite{LP}]
	An  \emph{alcove path} is a sequence of alcoves $(A_0, A_1, \ldots, A_m)$ such that
	$A_{j-1}$ and $A_j$ are adjacent, for $j=1,\ldots, m.$ We say that $(A_0, A_1, \ldots, A_m)$  
	is \emph{reduced} if it has minimal length among all alcove paths from $A_0$ to $A_m$.
\end{dfn}

Let $\lambda\in P$ be any weight, although dominant and anti-dominant $\lambda$  will play a special role.
Let $A_{\lambda}=A_{\circ}+\lambda$ be the translation of the fundamental alcove $A_{\circ}$ by $\lambda$.
	
\begin{dfn}[\cite{LP}]\label{deflch}
	The sequence of roots $\Gamma(\lambda)=(\beta_1, \beta_2, \dots, \beta_m)$ is called a
	\emph{$\lambda$-chain} if 
	\[	
		A_0=A_{\circ} \xrightarrow{-\beta_1}  A_1
		\xrightarrow{-\beta_2} \cdots 
		\xrightarrow{-\beta_m}  A_m=A_{-\lambda}
	\]
is a reduced alcove path.
\end{dfn}

A reduced alcove path $(A_0=A_{\circ},A_1,\ldots,A_m=A_{-\lambda})$ can be identified 
with the corresponding total order on the hyperplanes, to be called 
{\em $\lambda$-hyperplanes}, which separate $A_\circ$ from $A_{-\lambda}$. This total order is given by the sequence 
$H_{\beta_i,-l_i}$ for $i=1,\ldots,m$, where $H_{\beta_i,-l_i}$ contains the common wall of 
$A_{i-1}$ and $A_i$. Note that $\langle\lambda,\beta_i^\vee\rangle\ge0$, and that the integers $l_i$, called {\em heights}, have the following ranges:
\begin{equation}\label{ranges}0\le l_i\le\langle\lambda,\beta_i^\vee\rangle-1\;\;\mbox{if}\;\;\beta_i\in\Phi^+\,, \;\;\;\;\mbox{and}\;\;\;\;1\le l_i\le\langle\lambda,\beta_i^\vee\rangle\;\;\mbox{if}\;\;\beta_i\in\Phi^-\,.\end{equation}
 Note also that a $\lambda$-chain $(\beta_1, \ldots, \beta_m)$ determines 
the corresponding reduced alcove path, so we can identify them as well. 

\begin{rema}\label{remredword} {\rm 
A reduced alcove path corresponds to the choice of a reduced word for the affine Weyl group element sending $A_\circ$ to $A_{-\lambda}$ \cite[Lemma 5.3]{LP}. %For $\lambda$ dominant, another equivalent definition of an alcove path/$\lambda$-chain, based on a root interlacing condition which generalizes a similar condition characterizing reflection orderings, can be found in~\cite[Definition~4.1, Proposition~10.2]{LP1}.
}
\end{rema}

For dominant $\lambda$, we have a particular choice of a $\lambda$-chain, denoted by $\Gamma_{\rm lex}(\lambda)$, 
which we call the {\em lexicographic (lex) $\lambda$-chain} (see \cite[Prposition~6.7]{LP}).
For a $\lambda$-hyperplane $H_{\beta, -l}$, 
the rational number $l/\langle\lambda, \beta^\vee \rangle$ is called the {\em relative height}; by definition, the sequence of relative heights in the lex $\lambda$-chain is weakly increasing. 

The objects of the quantum alcove model are defined next; for examples, we refer to~\cite{lalgam,lnsumk2}. Compared with the original construction in~\cite{lalgam}, here we consider a generalization of this model, by letting $\lambda$ be any weight, as opposed to only a dominant weight; another aspect of the generalization is making the model depend on a fixed element  $w\in W$, such that the initial model corresponds to $w$ being the identity. In addition to $w$, we fix an arbitrary $\lambda$-chain $\Gamma(\lambda)=(\beta_1,\,\ldots,\,\beta_m)$, and let $r_i:=s_{\beta_i}$, $\widehat{r}_i:=s_{\beta_i,-l_i}$. 

\begin{dfn}[\cite{lalgam}]
\label{def:admissible}
	A subset 
	$A=\left\{ j_1 < \cdots < j_s \right\}$ of $[m]:=\{1,\ldots,m\}$ (possibly empty)
 	is a $w$-\emph{admissible subset} if
	we have the following directed path in $\QB(W)$:
	\begin{equation}
	\label{eqn:admissible}
	 \Pi(w,A):\;\;\;\;w\xrightarrow{|\beta_{j_1}|} w r_{{j_1}} 
	\xrightarrow{|\beta_{j_2}|}  wr_{{j_1}}r_{{j_2}} 
	\xrightarrow{|\beta_{j_3}|}  \cdots 
	\xrightarrow{|\beta_{j_s}|}  wr_{{j_1}}r_{{j_2}} \cdots r_{{j_s}}=:\wend(w,A)\,.
	\end{equation}
 	We let ${\mathcal A}(w,\Gamma(\lambda))$ be the collection of all $w$-admissible subsets of $[m]$.
\end{dfn}

We now associate several parameters with the pair $(w,A)$. 
The weight of $(w,A)$ is  
	\begin{equation}
	\label{defwta}
	\wt(w,A):=-w\rh_{{j_1}} \cdots 
	         \rh_{{j_s}}(-\lambda)\,.
	\end{equation}

Given the height sequence $(l_1,\ldots,l_m)$ above, we define 
the complementary height sequence $(\widetilde{l}_1,\ldots,\widetilde{l}_m)$ 
by $\widetilde{l}_i:=\langle\lambda,\beta_i^\vee\rangle-l_i$. 
Given $A=\{j_1<\cdots<j_s\}\in{\mathcal A}(w,\Gamma(\lambda))$, let 
%
\begin{equation*}
A^{-}:=\bigl\{j_i \in A \mid 
  wr_{{j_1}} \cdots r_{{j_{i-1}}} > 
 w r_{{j_1}} \cdots r_{{j_{i-1}}}r_{{j_{i}}}
\bigr\}\,;
\end{equation*}
%
in other words, we record the quantum steps 
in the path $\Pi(w,A)$ defined in \eqref{eqn:admissible}. 
Let
%
\begin{equation}
\label{defheight}
\dn(w,A):=\sum_{j\in A^-}|\beta_j|^\vee\in Q^{\vee,+}\,,\;\;\;\;\hgt(w,A):=\sum_{j\in A^-}\sgn(\beta_j)\widetilde{l}_j\,.
\end{equation}

\section{Chevalley formulas for the semi-infinite flag manifold}\label{csi}

Consider a simply-connected simple algebraic group $G$ over $\mathbb{C}$, with Borel subgroup $B = T N$, maximal torus $T$, and unipotent radical $N$. The full {\em semi-infinite flag manifold} $\mathbf{Q}_{G}^{\mathrm{rat}}$ is the reduced (ind-)scheme associated to $G(\bC(\!(z)\!))/(T\cdot N(\bC(\!(z)\!)))$; in this paper, we concentrate on its semi-infinite Schubert subvariety $\QQ := \QQ(e) \subset \mathbf{Q}_{G}^{\mathrm{rat}}$ corresponding to the identity element $e \in \Waf$, which we also call the semi-infinite flag manifold. The $T \times \mathbb{C}^*$-equivariant $K$-group $K_{T \times \mathbb{C}^*}(\QQ)$ of $\QQ$ has a (topological) $\mathbb{Z}[q, q^{-1}][P]$-basis of {\em semi-infinite Schubert classes}, and its multiplicative structure is determined by a {\em Chevalley formula}, which expresses the tensor product of a semi-infinite Schubert class with the class of a line bundle. In \cite{knsekt} and \cite{nospcf}, the Chevalley formulas were given in the cases of a dominant and an anti-dominant weight $\lambda$, respectively. These formulas were expressed in terms of the quantum LS path model. We will express them in terms of the quantum alcove model based on the lexicographic $\lambda$-chain. The goal is to generalize these formulas for an arbitrary weight $\lambda$, and we will also see that an arbitrary $\lambda$-chain can be used. Throughout this section, $W_J$ is the stabilizer of $\lambda$, and we use freely the notation in Section~\ref{backg}.

The $T \times \mathbb{C}^*$-equivariant $K$-group $K_{T \times \mathbb{C}^*}(\QQ)$ is the $\mathbb{Z}[q, q^{-1}][P]$-submodule of the (Iwahori-) equivariant $K$-group $K_{\mathbf{I} \rtimes \mathbb{C}^{*}}(\mathbf{Q}_{G}^{\mathrm{rat}})$ of $\mathbf{Q}_{G}^{\mathrm{rat}}$, introduced in \cite{knsekt}, consisting of all (possibly infinite) linear combinations of the classes $\OQG{x}$ of the structure sheaves of the semi-infinite Schubert varieties $\QQ(x) (\subset \QQ)$ with coefficients $a_{x} \in \mathbb{Z}[q, q^{-1}][P]$ for $x \in \Wafp = W \times Q^{\vee,+}$ such that the sum $\sum_{x \in \Wafp} \vert a_{x} \vert$ of the absolute values $\vert a_{x} \vert$ lies in $\mathbb{Z}_{\geq 0}(\!(q^{-1})\!)[P]$. Here $\mathbb{C}^*$ acts on $\QQ$ by loop rotation, and $\mathbb{Z}[P]$ is the group algebra of $P$, spanned by formal exponentials $\mathbf{e}^\mu$, for $\mu \in P$, with $\mathbf{e}^\mu \mathbf{e}^\nu = \mathbf{e}^{\mu+\nu}$; note that $\mathbb{Z}[P]$ is identified with the representation ring of $T$.
We also consider the $\mathbb{Z}[q, q^{-1}][P]$-submodule $K_{T \times \mathbb{C}^*}^{\prime}(\QQ)$ of $K_{T \times \mathbb{C}^*}(\QQ)$ consisting of all finite linear combinations of the classes $\OQG{x}$ with coefficients in $\mathbb{Z}[q, q^{-1}][P]$ for $x \in \Wafp$.
The $T$-equivariant $K$-groups of $\QQ$, denoted by $K_{T}(\QQ)$ and $K_{T}^{\prime}(\QQ)$, are obtained from the $K_{T \times \mathbb{C}^*}(\QQ)$ and $K_{T \times \mathbb{C}^*}^{\prime}(\QQ)$ above, respectively, by the specialization $q = 1$. Hence the Chevalley formulas in $K_T(\QQ)$ (for arbitrary weights) and $K_{T}^{\prime}(\QQ)$ (for anti-dominant weights) are obtained from the corresponding one in $K_{T \times \mathbb{C}^*}(\QQ)$ by setting $q=1$.
Note that $K_{T}(\QQ)$ turns out to be the $\bZ[P]$-module consisting of all (possibly infinite) linear combinations of the classes $\OQG{x}$, $x \in \Wafp$, with coefficients in $\bZ[P]$; also $K_{T}^{\prime}(\QQ)$ is the $\bZ[P]$-submodule of $K_{T}(\QQ)$ consisting of all finite linear combinations of the classes $\OQG{x}$, $x \in \Wafp$, with coefficients in $\bZ[P]$.



\subsection{Chevalley formulas for dominant and anti-dominant weights} 
We start with the Chevalley formula for dominant weights, which was derived in terms of semi-infinite LS paths in~\cite{knsekt}, and then restated in \cite[Corollary~C.3]{nospcf} in terms of quantum LS paths. 

Let $\lambda = \sum_{i \in I} \lambda_i \varpi_i$ be a dominant weight. We denote by $\overline{{\rm Par}(\lambda)}$ the set of $I$-tuples of partitions ${\bm{\chi}} = (\chi^{(i)})_{i \in I}$ such that $\chi^{(i)}$ is a partition of length at most $\lambda_i$ for all $i \in I$. For ${\bm{\chi}} = (\chi^{(i)})_{i \in I} \in \overline{{\rm Par}(\lambda)}$, we set $|{\bm{\chi}}| := \sum_{i \in I} |\chi^{(i)}|$, with $|\chi^{(i)}|$ the size of the partition $\chi^{(i)}$. Also set $\iota({\bm{\chi}}) := \sum_{i \in I} \chi^{(i)}_1 \alpha_i^{\vee} \in Q^{\vee,+}$, with $\chi^{(i)}_1$ the first part of the partition $\chi^{(i)}$.

\begin{thm}[\cite{knsekt,nospcf}]\label{chevdomqls} Let $x=wt_{\xi}\in \Wafp = W \times Q^{\vee,+}$. Then, in $K_{T \times \mathbb{C}^*}(\QQ)$, we have
\[\begin{split}&\OQGl{-w_\circ\lambda}\cdot\OQG{x}=\\[3mm]&\qquad=\sum_{\eta\in\QLS(\lambda)}\sum_{{\bm{\chi}}\in\overline{{\rm Par}(\lambda)}}q^{\Deg_w(\eta)-\langle\lambda,\xi\rangle-|{\bm{\chi}}|}\mathbf{e}^{\wt(\eta)}\OQG{\iota(\eta,w)t_{\xi+\xi(\eta,w)+\iota({\bm{\chi}})}}\,.\end{split}\]
\end{thm}

We now translate this formula in terms of the quantum alcove model for the lex $\lambda$-chain $\Gamma_{\rm lex}(\lambda)$. To this end, given $w\in W$, we construct a bijection $A\mapsto \eta$ between ${\mathcal A}(w,\Gamma_{\rm lex}(\lambda))$ and $\QLS(\lambda)$, for which several properties are then proved. 

In order to construct the forward map, let $A=\{j_1<\cdots<j_s\}$ be in ${\mathcal A}(w,\Gamma_{\rm lex}(\lambda))$. The corresponding heights are within the first range in~\eqref{ranges}. Consider the weakly increasing sequence of relative heights $h_i := l_{j_i}/ \langle \lambda,\beta_{j_i}^\vee \rangle \in [0,1) \cap \mathbb{Q}$ 
for $i=1,\ldots,s$.
%
%\begin{equation}\label{relh}
% h_i:=\frac{l_{j_i}}{\langle\lambda,\beta_{j_i}^\vee\rangle}\,\in[0,1)\cap{\mathbb{Q}}
% \,,\;\;\;\;\;i=1,\ldots,s\,.\end{equation}
%
Let $0<b_2<\cdots<b_t<1$ be the distinct nonzero values in the set $\{h_1,\ldots,h_s\}$, and let $b_1:=0$, $b_{t+1}:=1$. For $k=1,\ldots,t$, let $I_k:=\{1 \leq i \leq s \, \mid \, h_i=b_k\}$.%; these sets are all nonempty, except perhaps $I_1$. 

Recall the path $\Pi(w,A)$ in $\QB(W)$ defined in~\eqref{eqn:admissible}. We divide this path into subpaths corresponding to the sets $I_k$, and record the last element in each subpath; more precisely, for $k=0,\ldots,t$, we define the sequence of Weyl group elements
\[w_k:=w\prod_{i\in I_1\cup\cdots\cup I_k}^{\longrightarrow} r_{j_i}\,,\]
where the non-commutative product is taken in the increasing order of the indices $i$, and $w_0:=w$. For $k=1,\ldots,t$, let $\sigma_k:=\lfloor w_k\rfloor\in W^J$. We can now define the forward map as
\[(w,A)\,\mapsto\,\eta:=((b_1,b_2,\ldots,b_t,b_{t+1});\,(\sigma_1,\ldots,\sigma_t))\,.\]
We will verify below that the image is in $\QLS(\lambda)$. 

The inverse map is constructed using the quantum Deodhar lift and the related {\em shellability property} of the quantum Bruhat graph, due to Brenti-Fomin-Postnikov. 

\begin{prop}\label{qam2qls} The map $A\mapsto \eta$ constructed above is a bijection between ${\mathcal A}(w,\Gamma_{\rm lex}(\lambda))$ and $\QLS(\lambda)$. It maps the corresponding parameters in the following way:
\begin{multline*}\wt(w,A)=\wt(\eta),\: {\rm end}(w,A)=\iota(\eta,w),\\ \: {\rm down}(w,A)=\xi(\eta,w),\: -{\rm height}(w,A)=\Deg_w(\eta).\end{multline*}
\end{prop}

We translate the formula in Theorem~\ref{chevdomqls} to the quantum alcove model via Proposition~\ref{qam2qls}.

\begin{thm}\label{chevdomqam} Let $\lambda$ be a dominant weight, $\Gamma_{\rm lex}(\lambda)$ the lex $\lambda$-chain, and let $x=wt_{\xi}\in \Wafp$. Then, in $K_{T \times \mathbb{C}^*}(\QQ)$, we have
\[\begin{split}&\OQGl{-w_\circ\lambda}\cdot\OQG{x}=\\[3mm]&\qquad \sum_{A\in{\mathcal A}(w,\Gamma_{\rm lex}(\lambda))}\sum_{{\bm{\chi}}\in\overline{{\rm Par}(\lambda)}}q^{-{\rm height}(w,A)-\langle\lambda,\xi\rangle-|{\bm{\chi}}|}\mathbf{e}^{{\rm wt}(w,A)}\OQG{{\rm end}(w,A)t_{\xi+{\rm down}(w,A)+\iota({\bm{\chi}})}}\,.\end{split}\]
\end{thm}

A similar Chevalley formula for an anti-dominant weight $\lambda$ was derived in \cite[Theorem~1]{nospcf}, also in terms of quantum LS paths. Using a similar procedure to the one above, we translate it to the quantum alcove model, as stated in Theorem~\ref{corantidom}. We work with  the lex $\lambda$-chain $\Gamma_{\rm lex}(\lambda)$, defined as the reverse of the lex $(-\lambda)$-chain; note that the alcove path corresponding to the former (ending at $A_\circ-\lambda$) is the translation by $-\lambda$ of the alcove path corresponding to the latter (ending at $A_\circ+\lambda$). 
%%%%%%\revise{no need for YB moves}

\begin{thm}\label{corantidom} Let $\lambda$ be an anti-dominant weight, $\Gamma_{\rm lex}(\lambda)$ the lex $\lambda$-chain, and let $x=wt_{\xi}\in \Wafp$. Then, in $K_{T \times \mathbb{C}^*}^{\prime}(\QQ) \subset K_{T \times \mathbb{C}^*}(\QQ)$, we have
\[\begin{split}&\OQGl{-w_\circ\lambda}\cdot\OQG{x}=\\[3mm]&\qquad\sum_{A\in{\mathcal A}(w,\Gamma_{\rm lex}(\lambda))}(-1)^{|A|}q^{-{\rm height}(w,A)-\langle\lambda,\xi\rangle}\mathbf{e}^{{\rm wt}(w,A)}\OQG{{\rm end}(w,A)t_{\xi+{\rm down}(w,A)}}\,.
\end{split}\]
\end{thm}

\subsection{The Chevalley formula for an arbitrary weight}

We now exhibit the Chevalley formula for an arbitrary weight $\lambda=\sum_{i\in I}\lambda_i\varpi_i$. To state the formula, let $\overline{{\rm Par}(\lambda)}$ denote the set of $I$-tuples of partitions $\bm{\chi}=(\chi^{(i)})_{i\in I}$ such that $\chi^{(i)}$ is a partition of length at most $\max(\lambda_i,0)$.

\begin{thm}\label{genchev} Let $\lambda$ be an arbitrary weight, $\Gamma(\lambda)$ an arbitrary $\lambda$-chain, and let $x=wt_{\xi}\in \Wafp$. Then, in $K_{T\times{\mathbb C}^*}(\QQ)$, we have
\[\begin{split}&\OQGl{-w_\circ\lambda}\cdot\OQG{x}=\\[3mm]&\quad\!\!\!\!\sum_{A\in{\mathcal A}(w,\Gamma(\lambda))}\sum_{{\bm{\chi}}\in\overline{{\rm Par}(\lambda)}}(-1)^{n(A)}q^{-{\rm height}(w,A)-\langle\lambda,\xi\rangle-|{\bm{\chi}}|}\mathbf{e}^{{\rm wt}(w,A)}\OQG{{\rm end}(w,A)t_{\xi+{\rm down}(w,A)+\iota({\bm{\chi}})}}\,,
\end{split}\]
where $n(A)$, for $A=\{j_1<\cdots<j_s\}$, is the number of negative roots in $\{\beta_{j_1},\ldots,\beta_{j_s}\}$.
\end{thm}

\begin{exa}{\rm 
Assume that $\mathfrak{g}$ is of type $A_2$, and $\lambda=\varpi_1-\varpi_2$. 
%Assume that $\mathfrak{g}$ is of type $A_2$, and $\lambda=\omega_1-\omega_2$. 
Then, $\Gamma(\lambda):=(\alpha_{1},\,-\alpha_{2})$ is a $\lambda$-chain of roots.
% where $\alpha_{ij}:=\varepsilon_i-\varepsilon_j$. 
Assume that $w = s_{1}=s_{\alpha_{1}}$. In this case, we see that
${\mathcal A}(s_{1},\Gamma(\lambda))=\bigl\{\emptyset,\,\{1\},\,\{2\},\,\{1,2\}\bigr\}$, 
and we have the following table. 
%
\begin{equation*}
%
\begin{array}{c||c|c|c|c|c}
A & n(A) & {\rm height}(s_{1},A) & {\rm wt}(s_{1},A) & {\rm end}(s_{1},A) & {\rm down}(s_{1},A) \\ \hline\hline
\emptyset & 0 & 0 & s_{1}\lambda & s_{1} & 0 \\ \hline
\{1\} & 0 & 1 & \lambda & e & \alpha_{1}^{\vee} \\ \hline
\{2\} & 1 & 0 & s_{1}\lambda & s_{1}s_{2} & 0 \\ \hline
\{1,2\} & 1 & 1 & \lambda & s_{2} & \alpha_{1}^{\vee}
\end{array}
%
\end{equation*}
%
Also, we can identify $\overline{{\rm Par}(\lambda)}$ with $\mathbb{Z}_{\ge 0}$. 
Therefore, we obtain 
%
\begin{equation*}
\begin{split}
 & \OQGl{-w_\circ\lambda} \cdot \OQG{s_{1}t_{\xi}} = \\[3mm]
 & \hspace*{5mm} 
  \sum_{ m \in \mathbb{Z}_{\ge 0} } q^{ -\langle \lambda,\xi \rangle-m}
  \biggl\{%
  \underbrace{
  \mathbf{e}^{s_1\lambda}
  \OQG{ s_1 t_{\xi+m\alpha_{1}^{\vee}} }}_{A=\emptyset} 
  + 
  \underbrace{
  q^{-1} \mathbf{e}^{\lambda}
  \OQG{ t_{\xi+\alpha_{1}^{\vee}+m\alpha_{1}^{\vee}} }}_{A=\{1\}} \\
 & \hspace*{40mm}
   + 
   \underbrace{(-1)\mathbf{e}^{s_{1}\lambda}
   \OQG{ s_1s_2 t_{\xi+m\alpha_{1}^{\vee}} }}_{A=\{2\}}
   +
   \underbrace{ (-1)q^{-1} \mathbf{e}^{\lambda}
   \OQG{ s_2 t_{\xi+\alpha_{1}^{\vee}+m\alpha_{1}^{\vee}} }}_{A=\{1,2\}} \biggr\}. 
\end{split}
\end{equation*}}
\end{exa}

\section{The quantum $K$-theory of flag manifolds}\label{qkgb}

Y.-P. Lee defined the (small) {\em quantum $K$-theory} of a smooth projective variety $X$, denoted by $QK(X)$ \cite{leeqkt}. This is a deformation of the ordinary $K$-ring of $X$, analogous to the relation between quantum cohomology and ordinary cohomology. The deformed product is defined in terms of certain generalizations of {\em Gromov-Witten invariants} (i.e., the structure constants in quantum cohomology), called {\em quantum $K$-invariants of Gromov-Witten type}. 

In order to describe the (small) $T$-equivariant quantum $K$-algebra $QK_T(G/B)$, for the finite-dimensional flag manifold $G/B$, we associate a variable $Q_k$ to each simple coroot $\alpha_k^{\vee}$, 
and let $\bZ[Q]:=\bZ[Q_1,\dots,Q_r]$. 
Given $\xi=d_1 \alpha_1^\vee+\cdots+d_r\alpha_r^\vee$ in $Q^{\vee,+}$, let $Q^\xi:=Q_1^{d_1}\cdots Q_r^{d_r}$. 
Let $\bZ[P][Q]:=\bZ[P]\otimes_{{\mathbb Z}} \bZ[Q]$, where the group algebra $\mathbb{Z}[P]$ of $P$ was defined at the beginning of Section~\ref{csi}. 
By the finiteness theorem of Anderson-Chen-Tseng (\cite{ACT}), we can define $QK_T(G/B)$ to be the $\bZ[P][Q]$-module
$K_T(G/B) \otimes_{{\mathbb Z}[P]} \bZ[P][Q] \subset K_{T}(G/B) \otimes_{\mathbb{Z}[P]} \bZ[P][\![Q]\!]$, where $\bZ[P][\![Q]\!] := \bZ[P] \otimes_{\mathbb{Z}} \bZ[\![Q]\!]$.
The algebra $QK_T(G/B)$ has a $\mathbb{Z}[P][Q]$-basis given by the classes $[{\mathcal O}^w]$ of the structure sheaves of the (opposite) Schubert varieties $X^{w} \subset G/B$ of codimension $\ell(w)$, for $w \in W$. 

It is proved in \cite{kat1} (see also \cite{kat2}) that there exists a $\mathbb{Z}[P]$-module isomorphism from $QK_T(G/B)$ onto $K_{T}^{\prime}(\QQ)$ that respects the quantum multiplication in $QK_T(G/B)$ and the tensor product in $K_{T}^{\prime}(\QQ)$;
in particular, it respects the quantum multiplication with the class of the line bundle $[\mathcal{O}_{G/B}(-\varpi_{k})]$ and the tensor product with the class of the line bundle $\OQGl{w_{\circ} \varpi_{k}}$, for $k \in I$.
Here we remark that in order to translate the Chevalley formula in $K_{T}(\QQ)$ for fundamental weights into the one in the quantum $K$-theory of $G/B$, we need to consider $K_{T}(G/B) \otimes_{\mathbb{Z}[P]} \bZ[P][\![Q]\!]$;
for example, in type $A_{n-1}$, the tensor product in $K_{T}(\QQ)$ with the class $\OQGl{-w_{\circ} \varepsilon_{k}}$ for $1 \leq k \leq n$ corresponds to the quantum multiplication with the class $\frac{1}{1-Q_{k}}[\mathcal{O}_{G/B}(\varepsilon_{k})]$, where $\varepsilon_{k} := \varpi_{k} - \varpi_{k-1}$, with $\varpi_{0} := 0$, $\varpi_{n} := 0$, and $Q_{n} := 0$.
Also, note that the above isomorphism sends each (opposite) Schubert class $\mathbf{e}^{\mu} [{\mathcal O}^w] Q^{\xi}$ in $QK_T(G/B)$ to the corresponding semi-infinite Schubert class $\mathbf{e}^{-\mu} \OQG{wt_{\xi}}$ in $K_{T}^{\prime}(\QQ)$ for $w \in W$, $\xi \in Q^{\vee,+}$, and $\mu \in P$. These results and the formula in Theorem~\ref{corantidom} imply an important conjecture in~\cite{LP}: the Chevalley formula for $QK_T(G/B)$;
we also use the equality $[\mathcal{O}^{s_k}] = 1 - \mathbf{e}^{-\varpi_k}[\mathcal{O}_{G/B}(-\varpi_k)]$ in $K_{T}(G/B)$.

\begin{thm}\label{qkchev} Let $k \in I$, and fix a $(-\varpi_k)$-chain of roots $\Gamma(-\varpi_k)$. Then, in $QK_{T}(G/B)$, we have
\[\begin{split}&[{\mathcal O}^{s_k}]\cdot [{\mathcal O}^w] = \\[3mm]&
(1-\mathbf{e}^{w(\varpi_k) - \varpi_k})  [{\mathcal O}^w] +
\sum_{A\in{\mathcal A}(w,\Gamma(-\varpi_k))\setminus\{\emptyset\}}
(-1)^{|A|-1} \,Q^{{\rm{down}}(w,A)}\mathbf{e}^{-\varpi_k-{\rm{wt}}(w,A)} [{\mathcal O}^{{\rm end}(w,A)}].\end{split}\]
\end{thm}

Let us now turn to the type $A_{n-1}$ flag manifold $Fl_{n}=SL_{n}/B$ and its (non-equivariant) quantum $K$-theory $QK(Fl_{n})$. In~\cite{lamqgp}, the first author and Maeno defined the so-called {\em quantum Grothendieck polynomials}. According to~\cite[Theorem~6.4]{lamqgp}, whose proof is based on intricate combinatorics, the mentioned polynomials multiply precisely as stated by the above Chevalley formula; note that in the (non-equivariant) $K$-theory $K(G/B)$, the (opposite) Schubert class $[\mathcal{O}^{w}]$ is identical to the class of the structure sheaf of the Schubert variety $X_{w_{\circ} w} \subset G/B$ of codimension $\ell(w)$ for $w \in W$. As this formula determines the multiplicative structure of $QK(Fl_{n})$~\cite{BCMP}, we derive the following result, settling the main conjecture in~\cite{lamqgp}.

\begin{thm}\label{qgroth} The quantum Grothendieck polynomials represent Schubert classes in $QK(Fl_{n})$. 
\end{thm}

Given a {\em degree} $d=(d_1,\ldots, d_{n-1})$, let $N_{s_k,w}^{v,d}$ be the coefficient of $Q_1^{d_1}\cdots Q_{n-1}^{d_{n-1}}[{\mathcal O}^v]$ in the expansion of $[{\mathcal O}^{s_k}]\cdot [{\mathcal O}^w]$ in $QK(Fl_{n})$ for $k\in I= \{1, \ldots, n-1\}$. Based on Theorem~\ref{qkchev} and results on a charge statistic due to the first author, we proved the following theorem, which completely determines the Chevalley coefficients $N_{s_k,w}^{v,d}$.

\begin{thm}
For every $k,\,v$ and parabolic coset $\sigma W_{I\setminus \{k\}}$ not containing $v$, there are unique $d$ and $w\in \sigma W_{I\setminus \{k\}}$  (they can be constructed explicitly), such that $N_{s_k,w}^{v,d}=\pm 1$ (the sign is as in Theorem~{\rm \ref{qkchev}}). All the other coefficients $N_{s_k,w}^{v,d}$ are $0$. Moreover, in the expansion of $[{\mathcal O}^{s_k}]\cdot [{\mathcal O}^w]$ there is a minimum and a maximum degree (with respect to the componentwise order), which are constructed explicitly.  
\end{thm}

\vspace{-\baselineskip}
\section{The quantum $K$-theory of partial flag manifolds}\label{partial}

As an application of Theorem~\ref{qkchev}, we give a Chevalley formula for partial flag manifolds corresponding to minuscule weights in types $A$, $B$, $D$, and $E$;
also, in type $C$, we give a Chevalley formula for partial flag manifolds corresponding to all fundamental weights.

Let $k \in I$ be such that $\varpi_{k}$ is a minuscule fundamental weight in types $A$, $B$, $D$, and $E$; also, let $k \in I$ be arbitrary in type $C$.
Let $P_{J} \supset B$ be the maximal (standard) parabolic subgroup of $G$ associated to the subset $J := I \setminus \{ k \}$.
The $T$-equivariant quantum $K$-theory $QK_{T}(G/P_{J})$ of the partial flag manifold $G/P_{J}$ is defined as $K_{T}(G/P_{J}) \otimes_{\mathbb{Z}[P]} \mathbb{Z}[P][Q_{k}]$,
where $K_{T}(G/P_{J})$ is the $T$-equivariant $K$-theory of $G/P_{J}$, and $\mathbb{Z}[Q_{k}]$ is the polynomial ring in the single (Novikov) variable $Q_{k} = Q^{\alpha_{k}^{\vee}}$ corresponding to the simple coroot $\alpha_{k}^{\vee}$. The (opposite) Schubert classes $[\mathcal{O}_{J}^{y}]$, for $y \in W^{J}$, form a $\mathbb{Z}[P][Q_k]$-basis. 
It is proved in~\cite{kat2} that there exists a $\mathbb{Z}[P]$-module surjection $\Phi_{J}$ from $QK_{T}(G/B)$ to $QK_{T}(G/P_{J})$ such that $\Phi_{J}([\mathcal{O}^{w}]) = [\mathcal{O}_{J}^{\lfloor w \rfloor}]$ for each $w \in W$, and $\Phi_{J}(Q^{\xi}) = Q^{[\xi]^{J}}$ for each $\xi \in Q^{\vee,+}$, where $[\xi]^{J} := c_{k} \alpha_{k}^{\vee}$ for $\xi = \sum_{i \in I} c_{i} \alpha_{i}^{\vee} \in Q^{\vee,+}$.
Also, it is proved in \cite{kat2} that $\Phi_{J}([\mathcal{O}_{G/B}(- \varpi_{k})]) = [\mathcal{O}_{G/P_{J}}(- \varpi_{k})]$.
As is shown in \cite{BCMP}, the quantum multiplication in $QK_{T}(G/P_{J})$ is uniquely determined by its $\mathbb{Z}[P]$-module structure and the quantum multiplication with the class of the line bundle $[\mathcal{O}_{G/P_{J}}(- \varpi_{k})]$.
Therefore, the $\mathbb{Z}[P]$-module surjection $\Phi_{J}$ respects the quantum multiplications in $QK_{T}(G/B)$ and $QK_{T}(G/P_{J})$.

Based on the facts above, we obtain a formula for the quantum multiplication with $[\mathcal{O}_{G/P_{J}}(- \varpi_{k})]$ in $QK_{T}(G/P_{J})$ from Theorem~\ref{qkchev} in $QK_{T}(G/B)$ by applying $\Phi_{J}$;
this argument works for an arbitrary fundamental weight $\varpi_{k}$ of $G$ of any type.
However, upon applying $\Phi_{J}$, there are many terms to be cancelled in the corresponding formula in $QK_{T}(G/P_{J})$.
For a minuscule fundamental weight $\varpi_{k}$ of $G$ of type $A$, $B$, $D$, or $E$, and for an arbitrary fundamental weight $\varpi_{k}$ of $G$ of type $C$, we cancel out all these terms via a sign-reversing involution, and obtain a cancellation-free formula.
Below we give such a formula in type $C$, which is not included in \cite{BCMP}.
For each $k \in I$, consider a specific ($- \varpi_{k}$)-chain $\Gamma(k)$
whose initial segment $\Gamma^{1}(k)$ is of the following form:
\begin{equation*}
\varepsilon_{1} + \varepsilon_{2}; \varepsilon_{1} + \varepsilon_{3}, \varepsilon_{2} + \varepsilon_{3}; \ldots ; \varepsilon_{1} + \varepsilon_{k}, \varepsilon_{2} + \varepsilon_{k}, \ldots , \varepsilon_{k-1} + \varepsilon_{k}.
\end{equation*}
For a $w$-admissible subset $A \in \mathcal{A}(w, \Gamma(k))$, with $w \in W^{J}$, we set $A^{1} := A \cap \Gamma^{1}(k)$.

\begin{thm}\label{parabolic} Let $G$ be of type $C_{r}$, and $k \in I$ arbitrary. Then, for a given $w \in W^{J}$, we have the following cancellation-free formula in $QK_{T}(G/P_{J})$, where $\mathcal{A}_{\lessdot}(w, \Gamma(k))$ consists of all the $w$-admissible subsets $A$ with $\Pi(w, A)$ a saturated chain in Bruhat order:
\begin{equation*}
\begin{split}
[\mathcal{O}_{J}^{w}] \cdot [\mathcal{O}_{G/P_{J}}(- \varpi_{k})]
&= \sum_{A \in \mathcal{A}_{\lessdot}(w, \Gamma(k))} (-1)^{\vert A \vert} \mathbf{e}^{\mathrm{wt}(w, A)} [\mathcal{O}_{J}^{\mathrm{end}(w, A)}] \\
&- Q_{k} \sum_{\substack{A \in \mathcal{A}_{\lessdot}(w, \Gamma(k)) \\
\mathrm{end}(w, A^{1}) \geq \lfloor s_{\theta} \rfloor}}
(-1)^{\vert A \vert} \mathbf{e}^{\mathrm{wt}(w, A)} [\mathcal{O}_{J}^{\lfloor \mathrm{end}(w, A) s_{2 \varepsilon_{k}} \rfloor}]\,.
\end{split}
\end{equation*}
\end{thm}

\vspace{-\baselineskip}
%\bibliographystyle{plain}
%\bibliography{22Lenart.bib}
\printbibliography

\end{document}

\begin{thebibliography}{99}

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\end{thebibliography}
\end{document}
